Bruno Guillaume

Bio: Bruno Guillaume is an academic researcher from University of Paris-Sud. The author has contributed to research in topics: Normalization (statistics) & Typed lambda calculus. The author has an hindex of 2, co-authored 2 publications receiving 68 citations.

A λ-calculus with explicit weakening and explicit substitution

Abstract: Since Mellies showed that λσ (a calculus of explicit substitutions) does not preserve the strong normalization of the β-reduction, it has become a challenge to find a calculus satisfying the following properties: step-by-step simulation of the β-reduction, confluence on terms with metavariables, strong normalization of the calculus of substitutions and preservation of the strong normalization of the λ-calculus. We present here such a calculus. The main novelty of this calculus (given with de Bruijn indices) is the use of labels that represent updating functions and correspond to explicit weakening. A typed version is also presented.

The λ s e -calculus does not preserve strong normalisation

Abstract: Kamareddine, F., & Rios (1997) conjecture that the λse-calculus preserves the strong normalisation of the λ-calculus. We prove here that this conjecture is false.

Term Rewriting Systems

Abstract: In this chapter we will present the basic concepts of term rewriting that are needed in this book. More details on term rewriting, its applications, and related subjects can be found in the textbook of Baader and Nipkow [BN98]. Readers versed in German are also referred to the textbooks of Avenhaus [Ave95], Bundgen [Bun98], and Drosten [Dro89]. Moreover, there are several survey articles [HO80, DJ90, Klo92, Pla93] that can also be consulted.

Nominal rewriting

Abstract: Nominal rewriting is based on the observation that if we add support for @a-equivalence to first-order syntax using the nominal-set approach, then systems with binding, including higher-order reduction schemes such as @l-calculus beta-reduction, can be smoothly represented. Nominal rewriting maintains a strict distinction between variables of the object-language (atoms) and of the meta-language (variables or unknowns). Atoms may be bound by a special abstraction operation, but variables cannot be bound, giving the framework a pronounced first-order character, since substitution of terms for variables is not capture-avoiding. We show how good properties of first-order rewriting survive the extension, by giving an efficient rewriting algorithm, a critical pair lemma, and a confluence theorem for orthogonal systems.

Nominal rewriting systems

Abstract: We present a generalisation of first-order rewriting which allows us to deal with terms involving binding operations in an elegant and practical way. We use a nominal approach to binding, in which bound entities are explicitly named (rather than using a nameless syntax such as de Bruijn indices), yet we get a rewriting formalism which respects α-conversion and can be directly implemented. This is achieved by adapting to the rewriting framework the powerful techniques developed by Pitts et al. in the FreshML project.Nominal rewriting can be seen as higher-order rewriting with a first-order syntax and built-in α-conversion. We show that standard (first-order) rewriting is a particular case of nominal rewriting, and that very expressive higher-order systems such as Klop's Combinatory Reduction Systems can be easily defined as nominal rewriting systems. Finally we study confluence properties of nominal rewriting.

The theory of calculi with explicit substitutions revisited

Abstract: Calculi with explicit substitutions (ES) are widely used in different areas of computer science. Complex systems with ES were developed these last 15 years to capture the good computational behaviour of the original systems (with meta-level substitutions) they were implementing. In this paper we first survey previous work in the domain by pointing out the motivations and challenges that guided the development of such calculi. Then we use very simple technology to establish a general theory of explicit substitutions for the lambda-calculus which enjoys fundamental properties such as simulation of one-step beta-reduction, confluence on metaterms, preservation of beta-strong normalisation, strong normalisation of typed terms and full composition. The calculus also admits a natural translation into Linear Logic's proof-nets.

Resource operators for λ-calculus

Abstract: We present a simple term calculus with an explicit control of erasure and duplication of substitutions, enjoying a sound and complete correspondence with the intuitionistic fragment of Linear Logic's proof-nets. We show the operational behaviour of the calculus and some of its fundamental properties such as confluence, preservation of strong normalisation, strong normalisation of simply typed terms, step by step simulation of @b-reduction and full composition.